Question

solve for $x$ using the figure to the right.
$x=\square$ (simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Identify similar triangles

The large right triangle and the smaller right triangle containing side \(x\) are similar, so their corresponding sides are proportional. The hypotenuse of the large triangle is \(15 + 4 = 19\), the hypotenuse of the smaller triangle is \(15\), and the side corresponding to \(x\) in the large triangle is \(x\) (wait, correct proportionality: the side \(x\) corresponds to the segment of length 15, and the side of length 4 corresponds to the other leg of the small triangle, but more accurately, for similar right triangles formed by an altitude: if we let the altitude be \(x\), then \(x^2 = 15 \times 4\) (geometric mean theorem: in a right triangle, the altitude to the hypotenuse is the geometric mean of the segments it divides the hypotenuse into).

Step2: Calculate \(x^2\)

\(x^2 = 15 \times 4 = 60\)

Step3: Solve for \(x\)

Take the square root of both sides, since \(x\) is a length, we take the positive root:
\(x = \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}\)

Answer:

\(2\sqrt{15}\)